| G1 | G2 | G3 |
|---|---|---|
| 2 | -1.0 | -1.0 |
| 1 | -0.5 | -0.5 |
| 4 | -2.0 | -2.0 |
Analysis of Variance
Department of Educational Psychology
Agenda
1 Overview and Introduction
2 The F Distribution
3 The F Ratio
4 One-way ANOVA
5 Mean Comparisons in One-way ANOVA
6 Conclusion
Agenda
1 Overview and Introduction
2 The F Distribution
3 The F Ratio
4 One-way ANOVA
5 Mean Comparisons in One-way ANOVA
6 Conclusion
Agenda
1 Overview and Introduction
2 The F Distribution
3 The F Ratio
4 One-way ANOVA
5 Mean Comparisons in One-way ANOVA
6 Conclusion
\[ F = \frac{MS_{between}}{MS_{within}} \]
\[ F = \frac{MS_{between}}{MS_{within}} \]
\[ MS_{between} = \frac{SS_{between}}{df_{between}} \]
\[ df_{between} = k - 1 \]
\[ SS_{between} = \sum{[\frac{(s_j)^2}{n_j}] - \frac{(\sum{s_j})^2}{n}} \]
\[ SS_{total} = \sum{x^2 \cdot \frac{(\sum{x})^2}{n}} \]
\[ MS_{within} = \frac{SS_{within}}{df_{within}} \]
\[ df_{within} = n - k \]
\[ SS_{within} = SS_{total} - SS_{between} \]
Agenda
1 Overview and Introduction
2 The F Distribution
3 The F Ratio
4 One-way ANOVA
5 Mean Comparisons in One-way ANOVA
6 Conclusion
Much like the other tests, we need to be mindful of several assumptions that underline this statistical test
These assumptions are:
Of course, this is only the tip of the iceberg
One question that the ANOVA alone doesn't answer: which of the groups are different?
There is also the two-way ANOVA, which allows us to compare two or more independent variables and their combined and interacting impact on a single dependent variable in two-way Analysis of Variance.
Finally, there is the repeated-measures ANOVA
\[ \eta^2 = \frac{SS_{Between}}{SS_{Total}} \]
\[ \omega^2 = \frac{SS_{Between} - (k-1)(MS_{within})}{SS_{total} + MS_{within}} \]
Agenda
1 Overview and Introduction
2 The F Distribution
3 The F Ratio
4 One-way ANOVA
5 Mean Comparisons in One-way ANOVA
6 Conclusion
| G1 | G2 | G3 |
|---|---|---|
| 2 | -1.0 | -1.0 |
| 1 | -0.5 | -0.5 |
| 4 | -2.0 | -2.0 |
Agenda
1 Overview and Introduction
2 The F Distribution
3 The F Ratio
4 One-way ANOVA
5 Mean Comparisons in One-way ANOVA
6 Conclusion
We can think of a one-way ANOVA as being similar to an independent-samples t-test, but for more than 2 groups. Just note the differences in the null/alternative hypotheses setup
There are some important extensions and other applications of the ANOVA that we did not comprehensively cover here. However, they do still employ the F-distribution and have the underlying focus on comparing variances to make conclusions about differences between groups. We will cover those later on!
The F-distribution adds to the other practical distributions employed as part of hypothesis testing, such as the t-distribution and chi-squared distribution (which you learned about in EDPS-641). Just like those, it allows us to conclude whether a null hypothesis can be rejected or retained.
On top of performing a One-way ANOVA, we can also follow it up with various Mean Comparisons in One-way ANOVA! These help us determine where our differences exists between our groups, and can offer important insights on top of our one-way ANOVA results. We continue working through these in next lecture.
Module 4 Lecture - One-way ANOVA and Multiple Comparison Procedures || Analysis of Variance